1. Depends on the situation. In some cases you'd deal more damage with slime + bombardier. I'd suggest slime for quests where you don't have the element to match, or for multiple monster quests.
2. Just check the wiki, the information isn't given to you in-game.
If you don't want to check a quest list though, with some guessing you can figure out which quests you need. Usually it's the tougher monsters in the quest list. But just check the wiki if you don't plan on doing every quest ;)